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\begin{document}

\title{Experimental study of the strengthening effect of low amplitude loading
history for a P/M nickel-based superalloy FGH96}



\author[1]{Xiaoguang YANG}%
\author[1]{RenJie Jiang}%
\author[1]{Duoqi SHI}%
\author[1]{Muwei Cheng}%
\author[2]{Yantao Sun}%
\affil[1]{Beihang University School of Energy and Power Engineering}%
\affil[2]{AVIC Beijing Aeronautical Manufacturing Technology Research Institute}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
The influence of preloading on the residual fatigue life of nickel-based
superalloys under elevated temperatures was investigated experimentally.
A powder metallurgy nickel-based superalloy FGH96 was preloaded with
different number of cycles, and the residual fatigue life was tested
under subsequent high-amplitude loads. The test results show that the
fatigue life of the material was increased when the preloading cycle
number was within a specific range. At the same time, the fatigue life
of virgin specimens under high-amplitude loads shows little scatter,
which provides the possibility to consider the strengthening effect of
low-amplitude loads. A novel damage accumulation model was proposed to
incorporate the strengthening effect of low-amplitude loads into the
life prediction framework. The proposed model provides better life
prediction than some existing models. Finally, the proposed model was
validated using experimental data for various materials under the
low-high loading sequence.%
\end{abstract}%



\sloppy


\textbf{Experimental study of the strengthening effect of low amplitude
loading history} \textbf{for a P/M nickel-based superalloy FGH96}

\textbf{Renjie Jiang\textsuperscript{1} Xiaoguang
Yang\textsuperscript{1} Duoqi Shi\textsuperscript{1} Muwei
Cheng\textsuperscript{1} Yantao Sun\textsuperscript{2}}

\textsuperscript{1}School of Energy and Power Engineering, Beihang
University, Beijing, China\textsuperscript{2}Beijing Aeronautical
Engineering Technical Research Center, Beijing,
China\textbf{Correspondence}Xiaoguang Yang, School of Energy and Power
Engineering, Beihang University, Beijing 100191, China. Email:
yxg@buaa.edu.cn

\textbf{Abstract}

The influence of preloading on the residual fatigue life of nickel-based
superalloys under elevated temperatures was investigated experimentally.
A powder metallurgy nickel-based superalloy FGH96 was preloaded with
different number of cycles, and the residual fatigue life was tested
under subsequent high-amplitude loads. The test results show that the
fatigue life of the material was increased when the preloading cycle
number was within a specific range. At the same time, the fatigue life
of virgin specimens under high-amplitude loads shows little scatter,
which provides the possibility to consider the strengthening effect of
low-amplitude loads. A novel damage accumulation model was proposed to
incorporate the strengthening effect of low-amplitude loads into the
life prediction framework. The proposed model provides better life
prediction than some existing models. Finally, the proposed model was
validated using experimental data for various materials under the
low-high loading sequence.

\textbf{Keywords:} Fatigue life prediction; Loading sequence effects;
Cumulative fatigue damage; Ni-base superalloy

\textbf{NOMENCLATURE}

material parameter of the strengthening factor

material parameter of the strengthening factor

accumulative fatigue damage

accumulated damage after -level stress loading material parameter of the
strengthening factor

loading cycles under the stress level

the fatigue life of the virgin material under the stress level

the strengthening factor

stress function corresponding to the stress level of the Chaboche model

applied stress under the stress level

\textbf{1 INTRODUCTION}

Operating under variable fatigue loadings, the accumulation of fatigue
damage is one of the main reasons for the failure of structural
components. The loading sequence has been found to affect fatigue damage
accumulation significantly\textsuperscript{1--4}. The influence of
loading sequence on fatigue damage has been widely investigated, and
many nonlinear cumulative damage models have been
developed\textsuperscript{5--10}.

Depending on the relationship between fatigue damage and cycle life
fraction, existing cumulative damage models can be divided into linear
and nonlinear models. The Palmgren-Miner rule\textsuperscript{11} is the
earliest linear cumulative damage model, widely used due to its
simplicity. However, it has been found that Miner's rule ignores the
effect of loading sequence on fatigue damage\textsuperscript{1}; its
predictions tend to be non-conservative under high-low
(\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{1}\textgreater{}\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{2}) loading sequence
and conservative under low-high
(\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{1}\textless{}\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{2}) loading sequence.
Numerous nonlinear cumulative damage models have been developed to
overcome this shortcoming. Fatemi and
Yang\textsuperscript{8}comprehensively reviewed cumulative damage
models. In addition, a review of cumulative damage models for high-cycle
fatigue was published by Hectors and De Waele\textsuperscript{12}.
Nonlinear cumulative damage models are generally categorized into five
categories: (a) models based on the S-N curve\textsuperscript{13--15},
(b) models based on crack growth concepts\textsuperscript{16,17}, (c)
models based on continuum damage mechanics
(CDM)\textsuperscript{18--20}, (d) energy-based
theories\textsuperscript{21--23}, (e) material degradation-based
models\textsuperscript{24,25}. In the existing models, the nonlinear
continuous damage model developed by Chaboche\textsuperscript{18}(the
Chaboche model) describes the progressive deterioration process before
macroscopic crack initiation. The Chaboche model presents several
advantages: First, the material parameters can be easily determined from
S-N curves. Second, the mean stress is included in the damage rate
equation. Finally, the model can be extended to account for additional
loading conditions and material properties, such as multi-axial
loading\textsuperscript{26} and strain hardening.

In most cases, cumulative damage models assume residual fatigue strength
decreases with loading cycles. However, it has been found that fatigue
strength could be improved by low-amplitude loads below the fatigue
limit. The strengthening effect of pre-cyclic stress on the residual
fatigue strength was earliest investigated by
Gough\textsuperscript{27}et al. and was called the ''coaxing effect.''
Investigations\textsuperscript{28--32} on the strengthening effect have
indicated that the fatigue strength improvement is attributed to
beneficial variations in the material microstructure instead of a purely
statistical phenomenon. Nakagawa\textsuperscript{33} pointed out that
the strengthening effect is induced by work hardening and strain aging.
It has been indicated that the strengthening effect is related to the
stress level, the number of cycles, and load
sequence\textsuperscript{34--36}. In addition, the significance of the
strengthening effect varies with different
materials\textsuperscript{37}. Lu and Zheng
\textsuperscript{38,39}studied the strengthening effect of carbon steel.
They concluded that 75\% to 95\% of the fatigue limit has strengthening
effects, and the cycle range of significant strengthening effects is
from 2\selectlanguage{ngerman}×10\textsuperscript{5} to 4×10\textsuperscript{5}.

Some cumulative damage models have been developed to account for the
strengthening effect of low-amplitude loads below the fatigue limit. Zhu
et al.\textsuperscript{40} developed a modified Miner's rule using the
fuzzy set method. Zhang et al.\textsuperscript{41} introduced a
strengthening function into the Chaboche model to account for
low-amplitude loads. Zheng et al.\textsuperscript{42} established the
three-dimensional surface equations for fatigue life, strengthening
loads, and strengthening cycles. However, existing models only consider
loads below the fatigue limit. Investigations have revealed that cyclic
loads over the fatigue limit also exhibit strengthening
effects\textsuperscript{43,44}. Ishihara and
McEvily\textsuperscript{45}found that fatigue life can be improved by
cyclic loading. The tests show that the strengthening effect depends on
the preload stress level, i.e., the strengthening effect is most
significant when the preload stress level is slightly over the fatigue
limit. It is reported that the strengthening effect is attributed to the
internal stress induced by preloading\textsuperscript{44}. In fact, due
to the complexity of material properties and loading conditions, it
seems arbitrary to use the fatigue limit as a threshold to distinguish
between strengthening and damaging effects in all cases.

Aeroengine turbine discs are usually subjected to variable amplitude
cycles under elevated temperatures. These cycles can be divided into
primary and sub-cycles, and the amplitude of most sub-cycles is over the
fatigue limit. Ignoring the strengthening effects of these sub-cycles
may underestimate the turbine disc fatigue life. Therefore, for
nickel-based superalloys, which are turbine disc materials, it is
necessary to investigate the influence of pre-applied low-amplitude
loads on the residual fatigue life under elevated temperatures. The
experimental procedure of powder metallurgy nickel-based superalloy
FGH96 is described in Section 2. Section 3 presents the test results,
followed by a detailed discussion of the test results and fracture
surface observations. In Section 4, the existing fatigue cumulative
damage models are first briefly reviewed. Then a novel cumulative damage
model was developed by introducing a strengthening factor into the
Chaboche model. In Section 5, the predictive capabilities of the
proposed model and the typical models were compared using experimental
data for various materials. The main contributions and limitations of
the study are discussed in Section 6. Finally, the conclusions are
presented in Section 7.

\textbf{2 MATERIALS AND EXPERIMENTAL PROCEDURE}

\textbf{2.1 Materials}

FGH96 is a powder metallurgy nickel-based superalloy used to manufacture
turbine discs of advanced aero engines. This alloy is similar to Rene
88DT, a second-generation powder metallurgy superalloy from the United
States. The nominal composition of FGH96 is listed in Table 1. The 0.2\%
yield strength of FGH96 alloy at 600 \selectlanguage{english} is 1030 MPa. The fatigue limit at
600degC is 425 MPa and 763 MPa for R=-1 and R=0.05, respectively.

Table 1 The nominal composition of FGH96\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
Element & Cr & Co & W & Mo & Nb & Al\tabularnewline
\midrule
\endhead
Weight,\% & 15.50\textasciitilde{}16.50 & 12.50\textasciitilde{}13.50 &
3.80\textasciitilde{}4.20 & 3.80\textasciitilde{}4.20 &
0.60\textasciitilde{}0.80 & 1.96\textasciitilde{}2.30\tabularnewline
Element & Ti & C & Zr & B & Ni & Fe\tabularnewline
Weight,\% & 3.55-3.90 & 0.045\textasciitilde{}0.060 & 0.03-0.06 &
0.012-0.020 & Balanced & [?]0.50\tabularnewline
\bottomrule
\end{longtable}

\textbf{2.2 Experimental procedure}

Cylindrical specimens with a gauge diameter of 5 mm and a gauge length
of 25 mm were used in this study. All fatigue test specimens were cut
from the same turbine disc. These specimens were polished with fine
abrasive paper and then electropolished to minimize the effect of
residual stress on the surface.

Fatigue tests were stress-controlled and conducted on a Zwick Roell
resonance testing system. The loading frequency is 80Hz in the test. In
order to simulate turbine disc operation conditions, the maximum
stresses of high and low amplitude loads were designed to be the same
value, i.e., 1200 MPa, as shown in Fig 1. The test temperature was 600,
and the test was conducted in the air environment. It should be noted
that the entire test time is very short. It is known from the materials
handbook that creep deformation and creep damage caused by the mean
stress are very limited. Therefore, the creep effect in this test can be
neglected. The detailed test matrix is listed in Table 2.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

Fig 1 Loading spectrum of the experiment

Table 2 The test matrix of FGH96\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Specimen Number & \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a1}(MPa) & n\textsubscript{1} &
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a2}(MPa) & n\textsubscript{2}\tabularnewline
\midrule
\endhead
1 & 300 & Until failure & &\tabularnewline
2 & 300 & Until failure & &\tabularnewline
3 & & & 570 & Until failure\tabularnewline
4 & & & 570 & Until failure\tabularnewline
5 & & & 570 & Until failure\tabularnewline
6 & 300 & 171000 & 570 & Until failure\tabularnewline
7 & 300 & 341000 & 570 & Until failure\tabularnewline
8 & 300 & 341000 & 570 & Until failure\tabularnewline
9 & 300 & 521000 & 570 & Until failure\tabularnewline
\bottomrule
\end{longtable}

\textbf{3 EXPERIMENTAL RESULTS}

\textbf{3.1 Fatigue life results}

Detailed test results are listed in Table 3. The logarithmic means of
fatigue lives of specimens 1-2 and 3-5 were used as
N\textsubscript{f1}and N\textsubscript{f2}, respectively. It can be seen
that the residual cycle life was improved, especially when the preload
cycle life fraction is 0.5. It should be noted that the virgin specimens
3-5 show a narrow life scatter band under high-amplitude loading.
Therefore, the inherent dispersion of fatigue life is insufficient to
explain the increased residual life.

Table 3 Fatigue test results of FGH96\selectlanguage{english}
\begin{longtable}[]{@{}lllllllllll@{}}
\toprule
Specimen Number & N\textsubscript{f} & & & & & & & & &\tabularnewline
\midrule
\endhead
1 & 631279 & & & & & & & & &\tabularnewline
2 & 737330 & & & & & & & & &\tabularnewline
3 & 6845 & & & & & & & & &\tabularnewline
4 & 9517 & & & & & & & & &\tabularnewline
5 & 13810 & & & & & & & & &\tabularnewline
Specimen Number & n\textsubscript{1} & N\textsubscript{f1} &
N\textsubscript{f1} & n\textsubscript{1}/N\textsubscript{f1} &
n\textsubscript{1}/N\textsubscript{f1} & n\textsubscript{2} &
n\textsubscript{2} & N\textsubscript{f2} & N\textsubscript{f2} &
n\textsubscript{2}/N\textsubscript{f2}\tabularnewline
6 & 171000 & 682000 & 682000 & 0.25 & 0.25 & 16091 & 16091 & 9650 & 9650
& 1.667\tabularnewline
7 & 341000 & 682000 & 682000 & 0.5 & 0.5 & 50824 & 50824 & 9650 & 9650 &
5.265\tabularnewline
8 & 341000 & 682000 & 682000 & 0.5 & 0.5 & 46454 & 46454 & 9650 & 9650 &
4.812\tabularnewline
9 & 521000 & 682000 & 682000 & 0.75 & 0.75 & 13185 & 13185 & 9650 & 9650
& 1.366\tabularnewline
\bottomrule
\end{longtable}

\textbf{3.2 Result analysis}

FGH96 test results indicate that the residual fatigue life can be
increased by low-amplitude loading history. This phenomenon, often
referred to as the '' strengthening effect ''\textsuperscript{27}, can
be considered as a specific situation in variable amplitude loading. It
should be noted that the low-amplitude load in this experiment is higher
than the fatigue limit. At the same time, the strengthening effect is
usually considered to be induced by loads below the fatigue
limit\textsuperscript{38,39}. In fact, some existing tests also
demonstrated similar phenomena. Two-level test
results\textsuperscript{17,44--53} for various materials in low-high
sequence were collected and plotted in Fig 2. The low-amplitude loads in
these tests are all higher than the fatigue limit. The results of the
FGH96 test in this study are also plotted in Fig 2 for comparison.

As shown in Fig 2, test results with residual life fraction exceeding
unity can be regarded as exhibiting the strengthening effect. For
residual fatigue life fractions lower than unity, only the damaging
effect is exhibited\textsuperscript{39}. It is evident that the
strengthening effect exists only when the number of preload cycles is in
a specific range. All the preload cycle life fractions that exhibit
strengthening effects are less than 0.8. For FGH96 test results in this
study, 0.75N\textsubscript{f} is the maximum number of preloading cycles
that shows the strengthening effect, and the residual fatigue life is
nearly equal to the life of virgin material.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}

\end{center}
\end{figure}

Fig 2 Two-level test results for different
materials\textsuperscript{17,44--53} under the low-high loading sequence

The mechanism of material fatigue resistance improvement is still under
investigation\textsuperscript{54}. In Ishihara's
investigation\textsuperscript{45}, the strengthening effect is
considered to be due to the improvement of the crack growth threshold.
Observations show that a small number of loading cycles can make the
microstructure within the material more compact and uniform, hence
improving the crack growth resistance\textsuperscript{55}. There are
different explanations for other materials, including comprehensive
residual stress\textsuperscript{56}, microstructure
changes\textsuperscript{29}, and grain refinement\textsuperscript{57}.

It can be seen that the residual life fraction of FGH96 alloy is much
higher than other materials. The highest residual life fraction of FGH96
is near 5, while all other materials are less than 2. In Li et al.'s
research\textsuperscript{44}, the K417 alloy also exhibits significant
strengthening effects. Fig 2 shows that the residual fatigue life of
K417 was improved when the preload cycle life fraction was extremely
low. Li et al. found that low amplitude loading within a specific number
of cycles can reduce the strain response under subsequent loading. The
specimens with decreased strain response have a longer fatigue life than
the virgin specimens. In this study, FGH96 has similar test conditions
to K417, including elevated temperatures and high mean stresses.
Therefore, the strengthening effect of FGH96 may be attributed to the
cyclic hardening induced by previously cyclic loading.

In the microscope, the increase in fatigue strength may be attributed to
the strengthening of grain boundaries. Zheng et
al.\textsuperscript{54,58} pointed out that low amplitude cycles
increase the dislocation density near grain boundaries and form
dislocation walls. Hence the material's resistance against deformation
increases, and the fatigue strength is improved. When the number of
preloading cycles is small, only few hardened structures are formed in
the material. It can be seen that the residual fatigue life is only
slightly higher than the life of virgin material when the preload cycle
life fraction is 0.25. As the number of preloading cycles increases,
many dislocation tangles are generated, creating barriers to dislocation
movement. Thus the rate of plastic strain accumulation during subsequent
loading is reduced, which may result in longer fatigue life than virgin
materials\textsuperscript{59}. When the preload cycle lifetime fraction
nears 1, the damaging effect from dislocation accumulation offsets the
strengthening effect caused by cyclic hardening. Therefore, the residual
fatigue life will decrease to 0 with further increases in preload
cycles.

To investigate the effect of the number of preload cycles on the failure
mode, the fracture surfaces were observed. The fracture surfaces of
specimens 7-9 are shown in Fig 3. There are three distinct areas on the
macro fractography for each specimen. The area boundary is indicated by
dashed lines. Stage I represents the crack initiation and crack stable
propagation area. This area is brighter and smoother than the others.
Stage II represents the fast crack propagation area, and stage III
represents the final fast fracture area\textsuperscript{60}. The stage I
region of specimen 9 is smaller than specimens 7 and 8. This indicates
that a high number of preload cycles may result in a shorter crack
initiation and stable propagation time.

In addition, the stage I area of each specimen shows a semi-circle
shape, which suggests that the crack initiated from the surface. Fig
4(A), Fig 4(B), and Fig 4(C) illustrate the crack initiation areas of
specimens 7-9, respectively. Crystallographic facets can be observed in
the crack initiation area, which indicates that all cracks all initiated
at the twin grain boundary\textsuperscript{61,62}. Therefore, the
variation in residual cycle life is independent of the cracking mode.
Fig 4(D) shows the crack initiation area of specimen 3, which was tested
under constant high amplitude load. It can be seen that there are
multiple crystallographic facets on the specimen surface. Thus the
cracking mode of the specimen under constant load is similar to that of
the two-level load.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

Fig 3 The macro fractography of specimens 7-9. (A), Specimen 7,\emph{n}
\textsubscript{2}/\emph{N\textsubscript{f}} \textsubscript{2}=5.265.
(B), Specimen 8,\emph{n} \textsubscript{2}/\emph{N\textsubscript{f}}
\textsubscript{2}=4.812. (C), Specimen 9,\emph{n}
\textsubscript{2}/\emph{N\textsubscript{f}} \textsubscript{2}=1.366\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}

Fig 4 The Crack initiation area of specimens. (A), Specimen 7,\emph{n}
\textsubscript{2}/\emph{N\textsubscript{f}} \textsubscript{2}=5.265.
(B), Specimen 8,\emph{n} \textsubscript{2}/\emph{N\textsubscript{f}}
\textsubscript{2}=4.812. (C), Specimen 9,\emph{n}
\textsubscript{2}/\emph{N\textsubscript{f}} \textsubscript{2}=1.366.
(D), Specimen 3, constant load

\textbf{4 DAMAGE ACCUMULATION THEORY}

\textbf{4.1 Typical cumulative damage models}

In 1945, the hypothesis that fatigue damage is equal to the accumulated
life fraction was proposed by Miner based on Palmgren's work, referred
to as Miner's rule, expressed as:

where is the fatigue damage (the damage at failure is assumed to be
unity), and are the number of preload cycles and the number of cycles to
failure for the constant amplitude stress level , respectively. In the
two-level test, the residual life fraction can be expressed as:

where and are the preload cycle life fraction and the residual life
fraction, respectively. Due to its simplicity, Miner's rule has become
the standard method for fatigue design of metallic structures in
engineering\textsuperscript{63}. However, numerous
experiments\textsuperscript{46,64,65} revealed significant deviations
between the predictions of Miner's rule and the experimental results.
The main drawback of Miner's rule is its load sequence independence.
Various nonlinear cumulative damage models are proposed accounting for
loading sequence. Kwofie and Rahbar\textsuperscript{66} proposed the
concept of fatigue driving stress(FDS) based on the S-N curve, the
expression of FDS is:

where is fatigue driving stress, is the applied stress at the stress
level, and have their usual meaning, \emph{b} is the fatigue strength
exponent. The fatigue driving stress increases with cyclic loading, and
failure occurs when the fatigue driving stress reaches a critical value.
The critical fatigue driving stress is independent of the applied load.
Under variable amplitude loading, the fatigue driving stress is assumed
to remain constant as the load changes. The expression for the
cumulative damage based on the Kwofie and Rohbar model (K-R model for
short) is:

where is the number of failure cycles corresponding to the first stress
level. The residual life fraction in the two-level test can be expressed
as:

\par\null

From Equation , it can be seen that the K-R model takes into account the
loading sequence by multiplying the log-life ratio on Miner's rule. When
in the low-high sequence and the loading life fraction is small enough,
the residual life fraction predicted by Equation will be higher than
unity. This phenomenon implies that the residual fatigue strength is
improved due to low amplitude loading. Li et
al.\textsuperscript{44}modified the K-R model and proposed a new
nonlinear cumulative damage model that is expressed as:

where \emph{b} \textsubscript{1} and \emph{b} \textsubscript{2} are the
fatigue strength exponent of the first and second level stresses,
respectively. The Li model can account for cumulative damage under block
loading at different temperatures. In addition, this model allows
describing the significant increase of residual fatigue life due to
loading history.

Peng et al.\textsuperscript{67} developed a cumulative fatigue damage
model by combining the S-N curve and the material memory concept. The
S-N curve is assumed to translate and rotate clockwise with cyclic
loading. The fatigue damage is then described by the change in the S-N
curve slope and quantified by the material memory parameter. The Peng
model predicts the residual life fraction for the two-level test as:

\par\null

Peng model has a high sensitivity to loading history and can accurately
predict fatigue life under multi-level loading. In continuum damage
mechanics, the fatigue behavior of materials is regarded as a continuous
process in that damage accumulates progressively until macroscopic
cracks initiate. Chaboche and Lesne\textsuperscript{18} developed the
nonlinear continuous damage model based on continuum damage mechanics.
The differential relationship between the damage \emph{D} and the
applied cycles \emph{n} is expressed as:

\par\null

where is the maximum stress, is the mean stress, , and are material
constants. The function is expressed as:

\par\null

where is ultimate tensile strength, is the fatigue limit for fully
reversed conditions, is the material parameter determined by the damage
process. The Symbol is defined as if and if . For constant load loading,
the cumulative damage can be expressed as:

\par\null

Under variable amplitude loading, the accumulated damage from the
previous loading step can be transferred to the stress level, and the
equivalent cycle number that can yield the same damage is defined as:

\par\null

where is the equivalent cycle number, is Equation corresponding to the
loading level. Therefore, the cumulative damage under the loading level
is:

\par\null

Failure is considered to occur when equals unity. The residual life
fraction under two load levels can be obtained by substituting Equation
and Equation into Equation :

\par\null

\textbf{4.2 Proposed model}

The existing models are insufficient to describe the experimental
results of FGH96. The purpose of this study was to incorporate the
strengthening effect into the fatigue life evaluation framework. It
should be noted that it is difficult to account for multiple
micro-mechanisms when modeling the strengthening effect. Therefore, a
phenomenological model based on continuum damage mechanics was proposed
here.

It has been pointed out that the strengthening effect is closely related
to the preload stress level and the number of preload
cycles\textsuperscript{35}. According to the characteristics of
strengthening and damaging, the stress level of preload can be divided
into three regions\textsuperscript{39}, including the strengthening and
damaging region(\selectlanguage{greek}σ\selectlanguage{english}\textgreater{}\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{H}), the strengthening
region(\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{H}\textgreater{}\selectlanguage{greek}σ\selectlanguage{english}\textgreater{}\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{L}),
and the useless region (\selectlanguage{greek}σ\selectlanguage{english}\textless{}\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{L}).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image53/image53}

\end{center}
\end{figure}

\par\null

Fig 5 low amplitude load region in low-high sequence. (A), The
definition of load region. (B), Influence of preload on the residual
fatigue life

As shown in Fig 5(A), \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{H} and \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{L} are the
maximum and minimum strengthening stress, respectively; \selectlanguage{greek}σ \selectlanguage{english}is the stress
level of preload. Its influence on residual fatigue life can be omitted
for loads lower than the minimum strengthening stress. In the
strengthening region, the residual fatigue life can be improved when the
number of preload cycles is within the appropriate range. Several
investigations\textsuperscript{29,38,55} confirmed this phenomenon and
suggested that \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{H} is slightly below the fatigue limit.

For loads over the \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{H}, the residual fatigue life firstly
increases and then decreases with the number of preload cycles. This
opinion is supported by the test results exhibiting the strengthening
effect in Fig 2. In fact, for loads over the fatigue limit, the finite
fatigue life makes it inevitable that the damaging effect is dominated
when the number of cycles exceeds a specific value. The strengthening
effect is also affected by the stress level of the preload. Two-level
tests on various materials indicated that the strengthening effect
decreases with increasing stress level of the preload. Ishihara et
al.\textsuperscript{45} conducted two-level load tests on S45C steel
with preload stresses of 400 MPa and 500 MPa, respectively. The former
loading condition corresponds to a significant strengthening effect,
while the latter one can be ignored. The test results for DTD683
steel\textsuperscript{17} and Maraging steel\textsuperscript{49} also
show this trend. In Li et al.'s study\textsuperscript{44}, two-level
load tests with different stress levels were conducted on K417 alloy. It
was found that the residual fatigue life fraction decreased from 5.37 to
2.33 with increasing preload stress. Only the damaging effect can be
observed when the preload stress \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a1} is equal to or
higher than the subsequent stress \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a2}. The relationship
between\emph{n} \textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{1} and\emph{n}
\textsubscript{2}/\emph{N\textsubscript{f}} \textsubscript{2}must follow
Miner's rule when \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a1} equals \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a2}.

In the Chaboche model, the damage evolution in the low-high loading path
is shown in Fig 6. For materials exhibiting the strengthening effect, it
can be assumed that the fatigue resistance is improved after low
amplitude loading, i.e., the damage growth rate is reduced. The
expression of the damage rate can be modified from Equation to:

\par\null

where \emph{q} is a factor characterizing the strengthening effect and
can be referred to as the ``strengthening factor.''\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image55/image55}

\end{center}
\end{figure}

Fig 6 Schematic of the influence of the strengthing effect on the
evolution of damage in the low-high loading path

The strengthening factor \emph{q} is a function of the preload stress
level and the number of preload cycles. It has been deduced earlier that
the strengthening effect decreases as the first level of stress approach
the second level of stress. Therefore the ratio of two stress levels
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a1}/\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a2} can be used to characterize the
influence of the preload stress. Since the fatigue life of virgin
materials can implicitly indicate that the load is higher than the
fatigue limit, \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a1}/\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a2} was replaced by
the ratio of fatigue life corresponding to the first and second levels
of stress, i.e.,\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}.

In general, the influence of the stress level and the number of cycles
of the preload on the strengthening effect can be summarized in the
following four features:

(a) The strengthening effect can be ignored when the preload cycle life
fraction\emph{n} \textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{1}=0 or 1.

(b) The strengthening effect first increases and then decreases with the
preload cycle life fraction\emph{n}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{1}, and the
location of the ''turning point'' is related to the material.

(c) The strengthening effect increases with the life
ratio\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}between the
two stress levels.

(d) The strengthening effect is only observed under low-high sequence
loading, i.e.,\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{2}\textgreater{}1.

From Equation , it is clear that \emph{q} is always greater than or
equal to unity, and \emph{q} equals unity means that the strengthening
effect can be ignored. Thus the exponential function was used as the
basic form of the expression. Combining features (a), (b), (c), and (d),
the general form of \emph{q} was developed as:

\par\null

where \emph{A} , \emph{B} , and \emph{m} are material parameters, all
are greater than zero. (\emph{n}
\textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{1})\textsuperscript{A}*(1-\emph{n}
\textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{1})\textsuperscript{B}was used to represent features (a)
and (b). Parameters \emph{A} and\emph{B} affect the ``turning point'' in
the curve of \emph{q} . The preloading fatigue life fraction
corresponding to the maximum strengthening factor
\emph{q\textsubscript{max}} is\emph{A} /(\emph{A} +\emph{B} ). It still
has the same function if\emph{A} +\emph{B} is set as a constant. The
factor \emph{q} can be determined only by the parameters \emph{A} and
\emph{m} . It should be noted that the influence of this constant on
the\emph{q\textsubscript{max}} can be offset by taking a suitable value
for the parameter \emph{m} . Therefore, this constant can take an
arbitrary value when fitting the test results. For simplicity, the value
of\emph{A} +\emph{B} was taken as 2 here. Features (c) and (d) were
represented by in Equation . The parameter \emph{m} characterizes the
sensitivity of the strengthening factor \emph{q} to the life
ratio\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}.

Derived from Equation, the cumulative damage can be expressed as:

Then the residual life fraction is modified from Equation as follows:

\par\null

By substituting Equation into Equation, the expression for the residual
cycle life fraction considering the strengthening effect can be written
as:

\par\null

\textbf{5 MODEL VALIDATION}

\textbf{5.1 Life prediction for FGH96}

The residual cycle life fraction of FGH96 was predicted using the
proposed model. The experimental results and fitting curve are plotted
in Fig 7. The test results of specimen 6 and specimen 7 were used for
fitting to obtain the values of the parameters. The model
parameters\emph{A} and \emph{m} are 1 and 1.46, respectively. The
proposed model provides an appropriate description of the relationship
between the residual cycle life fraction and the preload cycle life
fraction. Fig 8 compares the experimental results and the predicted
lives of existing models\textsuperscript{11,44,66,67}. Here only
two-level test results (specimens 6-9) are included in Fig 8. It is
evident that the proposed model provides an acceptable prediction of the
residual life. All predictions using the proposed model fall into the
error band of 2. The existing models are insufficient to describe the
experimental results. The Peng model overestimates residual life, while
other models underestimate it.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image61/image61}

\end{center}
\end{figure}

Fig 7 FGH96 test results and fitting curve of the proposed model\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image62/image62}

\end{center}
\end{figure}

Fig 8 FGH96 test results versus the predicted lives using the proposed
and existing models\textsuperscript{11,44,66,67}

\textbf{5.2 Model validation for other materials}

Test results for other materials were collected to validate the proposed
model, including Ti-6Al-4V\textsuperscript{47},
Al-2024-T42\textsuperscript{48}, 30CrMnSiA\textsuperscript{52},
30NiCrMoV12\textsuperscript{46}, Maraging steel\textsuperscript{49},
GS61\textsuperscript{53}, ENAW6005\textsuperscript{51},
P355NL1\textsuperscript{50}, S45C\textsuperscript{45}, and
DTD683\textsuperscript{17}. To illustrate the model's applicability to
multiple materials, a set of universal parameters (\emph{A} =1,\emph{m}
=0.45) instead of material-dependent parameters was used to predict the
residual life. The equation is written as follows:

\par\null

As shown in Fig 9, these test results\textsuperscript{17,45--53} were
divided into four groups according to the value
of\emph{N\textsubscript{f}} \textsubscript{1}/\emph{N\textsubscript{f}}
\textsubscript{2}. For the test results with low life ratios(e.g.,
1\textless{}\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}[?]5), the
residual cycle life fraction is almost all less than unity and decreases
continuously with the preload cycle life fraction. For test results with
higher life ratios, it is evident that the residual cycle life fraction
is increased within a range of preload cycle life fractions. The
proposed model's predictions for each data group are also plotted in the
figure. The\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}used in the
proposed model is the maximum\emph{N\textsubscript{f}}
\textsubscript{1}/\emph{N\textsubscript{f}} \textsubscript{2}for each
group. It can be seen that the proposed model provides an appropriate
description of the data for different ranges of life ratios.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image64/image64}

\end{center}
\end{figure}

Fig 9 The experimental lives\textsuperscript{17,45--53} versus the
prediction results of the proposed model for different life ratios

The residual life was predicted using the proposed and typical
cumulative damage models. The comparisons between the experimental
results and the predicted life are illustrated in Fig 10. Most
predictions of the proposed model fall into the error band of 1.5, and
all fall within the error band of 2(Fig 10(A)). The predicted lives of
the K-R model\textsuperscript{66}, Miner's rule, and Li
model\textsuperscript{44} tend to be lower than the experimental
results(Fig 10(B), (C), (D)), especially for DTD683\textsuperscript{6},
S45C\textsuperscript{29}, and Maraging Steel\textsuperscript{49}, the
strengthening effect of the above three materials is significant(Fig 2).

In addition, the proposed model was compared with the typical cumulative
damage models using the fatigue life prediction error
model\textsuperscript{68} that is expressed as:

where \emph{P} \textsubscript{error} is the life prediction error in
logarithmic form, \emph{N\textsubscript{f}} \textsubscript{t}
and\emph{N\textsubscript{f}} \textsubscript{p} are the experimental and
predicted fatigue life. The mean and standard deviation of the
prediction errors were calculated to compare the predictive ability of
the models, and these values are listed in Table 4. The proposed model
provides better predictions compared to other models.

The accuracy of life prediction can be further improved by using
material-dependent parameters. However, the material-dependent
parameters \emph{A} and \emph{m} need to be obtained from experimental
data, which limits the application of the proposed model. Therefore it
is necessary to illustrate that universal parameters can provide
sufficient prediction accuracy for the collected data. Test results of
three materials were selected to compare prediction accuracy. Here tests
for each material contain more than three life ratios. The root mean
square(\emph{RMS} ) of the prediction error was used to compare the
prediction accuracy of different parameters:

where \emph{n} is the number of predictive lives. The \emph{RMS} of the
prediction errors is listed in Table 5. It is evident that the
deviations between the predictions using different parameters are
limited, and the universal parameters can provide enough accurate
predictions.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image67/image67}

\end{center}
\end{figure}

Fig 10 The predicted lives versus test results of different
materials\textsuperscript{17,45--53}. (A), Proposed model. (B), K-R
model\textsuperscript{66}. (C), Miner's rule\textsuperscript{11}. (D),
Li model\textsuperscript{44}. (E), Peng model\textsuperscript{67}

Table 4 Statistical analysis of model prediction errors\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Statistical Parameters & Proposed Model & K-R Model\textsuperscript{66}
& Miner's Rule\textsuperscript{1} & Li Model\textsuperscript{44} & Peng
Model\textsuperscript{67}\tabularnewline
\midrule
\endhead
Mean & 0.018 & -0.125 & -0.221 & -0.099 & 0.254\tabularnewline
Standard deviation & 0.113 & 0.177 & 0.188 & 0.254 &
0.283\tabularnewline
\bottomrule
\end{longtable}

Table 5 RMS of the prediction errors calculated using different
parameters\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Material\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
The universal parameters\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
The material-dependent parameters\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
30NiCrMoV12\textsuperscript{46}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=0.450,=1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.200\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=2.012,=1.953\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.192\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Maraging Steel\textsuperscript{49}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=0.450,=1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.146\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=0.056,=1.980\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.082\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
DTD683\textsuperscript{17}\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=0.450,=1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.088\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
=0.115,=0.192\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.083\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\textbf{6 DISCUSSIONS}

The strengthening effect has been observed in many investigations.
However, the existing fatigue life prediction models are mainly
concerned with the strengthening effect caused by loads lower than the
fatigue limit\textsuperscript{41}. There is limited attention to loads
higher than the fatigue limit. In this paper, we conducted two-level
load tests in the low-high sequence. Then a new fatigue damage model
based on the Chaboche model was developed to take into account the
strengthening effect. The main conclusion of this paper is the effect of
preloading cycle life fraction and load magnitude on the strengthening
effect. It's consistent with the test results of various materials in
the literature\textsuperscript{44,45,49}.

A limitation of the model is that only considering the preload
conditions. The mechanism that yields the strengthening effect is not
incorporated. In fact, there is still lack of a widely accepted
explanation for the strengthening effect\textsuperscript{54,58}. In many
cases, the pre-applied low amplitude loads only cause damaging rather
than strengthening\textsuperscript{46,47}. Hence the increase in cycle
life should be considered as a special case when specific materials and
loading conditions are combined. The model is suitable for describing
the phenomenon instead of predicting whether it occurs. In addition,
most of the tests were conducted under two-level loading. The
applicability of the model to multi-level loads needs further
validation.

In general, our work incorporated the strengthening effect into the
fatigue damage accumulation theory. Although the mechanism was not fully
understood, this study illustrated that the strengthening effect is
independent of the crack initiation mechanism. The tests in this study
showed that the strengthening effect is significant in specific
situations. For future research, a full understanding of the mechanism
will be valuable for cycle life design.

\textbf{7 CONCLUSIONS}

This paper investigated the strengthening effect of low amplitude loads
over the fatigue limit on residual cycle life in two-level loading.
Two-level load tests with different preload cycles under elevated
temperatures were conducted on FGH96 smooth specimens. The results show
that the fatigue life of the specimen was increased after subjecting
low-amplitude loading. Then test data from various materials were
collected to investigate the influence of the preload cycle life
fraction and the stress level of the preloading on residual cycle life.
Finally, a novel cumulative damage model based on continuum damage
mechanics was proposed to consider the strengthening effect under
low-high loading sequences. The conclusions are as follows:

\begin{enumerate}
\tightlist
\item
  In the low-high loading sequence, the residual fatigue life of FGH96
  under high-amplitude loads can be higher than the life of virgin
  specimens. The residual life first increases and then decreases with
  preload cycles.
\item
  It is assumed that the strengthening effect decreases with the stress
  level of the pre-applied load, and this assumption was verified on
  test data for various materials. For many materials, the strengthening
  effect increases with the ratio of fatigue life corresponding to
  preloading and subsequent stresses.
\item
  By introducing a strengthening factor into the Chaboche model, a novel
  cumulative damage model was proposed to take into account the
  strengthening effect. Compared to existing models, the proposed model
  provides a better description of the experimental results of FGH96. In
  addition, for the collected experimental results of various materials,
  the proposed model can provide accurate predictions using a set of
  universal parameters instead of material-dependent parameters.
\end{enumerate}

\textbf{ACKNOWLEDGMENTS}

This work is supported by the National Science and Technology Major
Project of China (J2019-IV-0017-0085) and National Natural Science
Foundation of China (12172021).

\textbf{DATA AVAILABILITY STATEMENT}

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

\textbf{CONFLICT OF INTEREST}

The authors declare that they have no conflict of interest.

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